James;

An example from Wiki:

For example, consider f(x) = 2x/(x + 1).f(100) = 1.9802 f(1000) = 1.9980 f(10000) = 1.9998As x becomes extremely large, the value of f(x) approaches 2, and the value of f(x) can be made as close to 2 as one could wish just by picking x sufficiently large. In this case, we say that the limit of f(x) as x approaches infinity is 2.In words:

The limit of f(x)=2, as x →∞.

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There are two different things being considered, f(x) and limit. The value 2 is assigned to the limit, not f(x).

Whether sequences converge or diverge or become an equality, depends on the f(x).

Specifically to the subject of inverse integers 1/n, if n is not a factor of 10 (2 or 5), a repeating decimal occurs.

In the world of pure mathematics, the f(x) for these sequences do not equal their assigned limits.

In the world of applied mathematics, these sequences are truncated for practical/convenient reasons.

If you insist on doing mathematics without using the concept of a limit, you will be severely limiting yourself.

Here is an example of using a limit as it relates to calculus.

Given, a continuous curve containing the points p, q', q.

The secant is the straight line v from a fixed point p to a variable point q.

As v rotates clockwise, it intersects the curve at a point q' closer to p.

The slope of v is m=y/x, the coordinates of its end point.

As q moves closer to p, the limit of the angle corresponding to m is theta.

When q coincides with p, v becomes the tangent for the curve at p.